Properties of $\inf_{m \geq n}f_m$ and $\sup_{m \geq n}f_m$ for a convergent sequence $f_n$

Question

Let $f_n$ be a sequence of real-valued functions converging to $f$. Define $g_n = \inf_{m \geq n} f_m$ and $h_n = \sup_{m \geq n} f_m$.

Where can I find properties of $g_n$ and $h_n$ (such as convergence results in function spaces and increasing/decreasing properties)? looking for a reference.